Programming Languages for Pre-Mechanical Calculating Tools

89 Pages·2012·2.83 MB·English

by  Baptiste Mélès

Introduction
Abacus
Counting-rods
Conclusion
Programming Languages
for Pre-Mechanical Calculating Tools
Baptiste Mélès 梅乐思
ArchivesHenriPoincaré–UniversitédeLorraine(Nancy南锡,France)
“Cultures of Mathematics and Logic”,
Guangzhou 广州, China, November 2012
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Introduction
“Manisananimalwhothinkswithhisfingers”
Abacus
Historyofmathematicsandcomputerscience
Counting-rods
Anexampleofstructuralanthropology: percussiontools
Conclusion
“Man is an animal who thinks with his fingers”
“As for the algebra that Chinese developped by applying it first to
the resolution of elementary problems of geometry, it can not be
dissociated from an ability to manipulations which is unequally
cultivated in the different civilizations. As said Marcel Mauss,
quoting Maurice Halbwachs: “Man is an animal who thinks with
his fingers” (« L’homme est un animal qui pense avec ses
doigts »), a beautiful maxim which warns us against too
intellectualist approaches, and which Chinese mathematicians would
probably have approved. The history of a civilization implies at
once socio-political structures, notions, mental attitudes,
behaviours, gestures, and everything that the ethnologists name
techniques of the body (« techniques du corps »).”
— Jacques Gernet, L’Intelligence de la Chine, “Histoire sociale et
intellectuelle”, p. 254.
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Introduction
“Manisananimalwhothinkswithhisfingers”
Abacus
Historyofmathematicsandcomputerscience
Counting-rods
Anexampleofstructuralanthropology: percussiontools
Conclusion
Question
Initial question: What functions can be computed with
different kinds of non-mechanical calculating tools?
Examples of calculating tools: abacus, Chinese counting-rods,
logarithm tables, slide rule, counting on paper...
Examples of limitations:
logarithm tables, slide rule: no addition;
abacus: no resolution of systems of n linear equations? no
logarithm?
3/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools
Introduction
“Manisananimalwhothinkswithhisfingers”
Abacus
Historyofmathematicsandcomputerscience
Counting-rods
Anexampleofstructuralanthropology: percussiontools
Conclusion
Question
Initial question: What functions can be computed with
different kinds of non-mechanical calculating tools?
Examples of calculating tools: abacus, Chinese counting-rods,
logarithm tables, slide rule, counting on paper...
Examples of limitations:
logarithm tables, slide rule: no addition;
abacus: no resolution of systems of n linear equations? no
logarithm?
3/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools
Introduction
“Manisananimalwhothinkswithhisfingers”
Abacus
Historyofmathematicsandcomputerscience
Counting-rods
Anexampleofstructuralanthropology: percussiontools
Conclusion
Question
Initial question: What functions can be computed with
different kinds of non-mechanical calculating tools?
Examples of calculating tools: abacus, Chinese counting-rods,
logarithm tables, slide rule, counting on paper...
Examples of limitations:
logarithm tables, slide rule: no addition;
abacus: no resolution of systems of n linear equations? no
logarithm?
3/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools
Introduction
“Manisananimalwhothinkswithhisfingers”
Abacus
Historyofmathematicsandcomputerscience
Counting-rods
Anexampleofstructuralanthropology: percussiontools
Conclusion
Classical answers
Two possible answers:
A posteriori (history of mathematics): “just read your classics”.
Counting-rods: Sunzi 孫子 (ca. 300), Sunzi Suanjing 孫子算
經;
Abacus: Cheng Dawei 程大位 (1533–1606), Suanfa Tongzong
算法統宗 (1592).
A priori (computer science): “just make a machine”.
Alan Turing (1912–1954), “On Computable Numbers” (1936):
the definition of “Turing machines” begins with the behaviour
of a human “computer” (i.e. the calculating man);
Joachim Lambek (born 1922), “How to Program an Infinite
Abacus” (1961).
4/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools
Introduction
“Manisananimalwhothinkswithhisfingers”
Abacus
Historyofmathematicsandcomputerscience
Counting-rods
Anexampleofstructuralanthropology: percussiontools
Conclusion
Classical answers
Two possible answers:
A posteriori (history of mathematics): “just read your classics”.
Counting-rods: Sunzi 孫子 (ca. 300), Sunzi Suanjing 孫子算
經;
Abacus: Cheng Dawei 程大位 (1533–1606), Suanfa Tongzong
算法統宗 (1592).
A priori (computer science): “just make a machine”.
Alan Turing (1912–1954), “On Computable Numbers” (1936):
the definition of “Turing machines” begins with the behaviour
of a human “computer” (i.e. the calculating man);
Joachim Lambek (born 1922), “How to Program an Infinite
Abacus” (1961).
4/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools
Introduction
“Manisananimalwhothinkswithhisfingers”
Abacus
Historyofmathematicsandcomputerscience
Counting-rods
Anexampleofstructuralanthropology: percussiontools
Conclusion
Classical answers
Two possible answers:
A posteriori (history of mathematics): “just read your classics”.
Counting-rods: Sunzi 孫子 (ca. 300), Sunzi Suanjing 孫子算
經;
Abacus: Cheng Dawei 程大位 (1533–1606), Suanfa Tongzong
算法統宗 (1592).
A priori (computer science): “just make a machine”.
Alan Turing (1912–1954), “On Computable Numbers” (1936):
the definition of “Turing machines” begins with the behaviour
of a human “computer” (i.e. the calculating man);
Joachim Lambek (born 1922), “How to Program an Infinite
Abacus” (1961).
4/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools
Introduction
“Manisananimalwhothinkswithhisfingers”
Abacus
Historyofmathematicsandcomputerscience
Counting-rods
Anexampleofstructuralanthropology: percussiontools
Conclusion
Limits
A posteriori (history of mathematics): “just read your classics”.
But the classics only describe what has been done, not
necessarily what can be done.
A priori (computer science): “just make a machine”.
But we lose the anthropological side of the problem. Idealized
computers are not computing men. (And infinite abaci do not
exist.)
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Introduction
“Manisananimalwhothinkswithhisfingers”
Abacus
Historyofmathematicsandcomputerscience
Counting-rods
Anexampleofstructuralanthropology: percussiontools
Conclusion
Structural anthropology and technology
To answer our question, we would need to mix those two
approches — anthropology and a priori science. But is it
possible?
Yes, it seems to be! There is at least one example. The idea
would be to have something like that:
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